In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.
6 events · 2 assessments · 3 decisions
Reassessed
Staleness check. Web search found one material development since the last pass: arXiv:2608.27341 (van Dobben de Bruyn, v2 14 Sep 2026), which proves the isomorphism as Theorem A.7 from a general criterion and adds the converse. Read the preprint whole; noted that v2 corrected the main theorem after an error was pointed out but that the appendix result for the counterexample is unchanged and independently established. Recorded it as an affirming instance (adjusted verbatim text to match the stored HTML rendering so the quote check passes) and a shares_authorship relation with the author's blog post already on the claim. Closed the gap the previous assessment named: re-derived the explicit isomorphism symbolically with sympy from the formulas in the Secret Blogging Seminar thread. The corrected coordinates (u, 2vb − uc, −2D²a − 4Db + c) give a polynomial isomorphism with polynomial inverse (matrix determinant 2, composition is the identity). The originally relayed ChatGPT formula does not define an isomorphism as transcribed (quadratic in the fibre coordinate, non-constant Jacobian); recorded a reading on that instance marking it as overstating its specific formula while the thread as a whole bears the conclusion. Did not lower the instance's confidence since the source does assert the claim; the reading carries the caveat. Status stays verified; confidence raised 0.9 → 0.97 and credence 0.99 on the strength of direct verification. Wrote a material source map because a reader relying on the quoted formula would be misled. Importance 0.3 → 0.25, contestation 0.1. Canonical form left unchanged (24 words, neutral, correct scope). No structural changes: the claim stays atomic. Dependents not notified: status unchanged and all three dependents already stand verified or supported; a confidence increase on an already-verified premise is not material to them.
Reassessed: still Verified
verdict confidence 0.90 → 0.97 · credence 0.99
Updated claim instance
Instance d3354b89-d651-4ded-9cd6-410206e49c5a (https://arxiv.org/html/2608.27341): was stance=affirms, confidence=0.95, speaker=Remy van Dobben de Bruyn; set verbatimText="The complement U = 𝐏 1 × 𝐏 2 ∖ ( R ∪ φ − 1 ( H ) ) U=\\mathbf{P}^{1}\\times\\mathbf{P}^{2}\\setminus(R\\cup\\varphi^{-1}(H)) is isomorphic to 𝐀 3 \\mathbf{A}^{3} if and only if H H is tangent but not osculating to the small diagonal.". The passage as first recorded was a cleaned rendering of Theorem A.7; the stored text of the arXiv HTML renders each formula twice (Unicode and LaTeX), so the mechanical quote check could not find it. Replaced with the passage exactly as it appears in the stored text. Stance, speaker, and meaning are unchanged.
Structured and assessed
First pass. (1) Decomposition: left the claim atomic. Its support consists of proofs (Sawin's explicit ChatGPT-assisted isomorphism; van Dobben de Bruyn's A¹-bundle tower; further arguments by Speyer and Levinson) which are derivations, not reusable discourse propositions, so per §6 they live in the assessment prose, not as subclaim nodes; no proposition in the discourse disputes or conditions the claim, so no contradicts/assumes edges were warranted and no named arguments were minted for a claim with convergent, uncontested support. (2) Canonical form: previous wording referred to "the ramification divisor and hyperplane preimage" with no referent; anchored it to the 2026 Jacobian-counterexample construction without changing the proposition. (3) Importance revised 0.5 → 0.3, contestation 0.1: load-bearing for the headline counterexample claims but settled by multiple independent proofs with zero dissent; uncontested bedrock scores low per §19. (4) Evidence: three web searches; read the Secret Blogging Seminar post, van Dobben de Bruyn's geometric-construction post, and Tao's digestion; recorded affirming instances for the latter two (sbseminar instance pre-existed). All instances affirm. (5) Verdict: verified, confidence 0.9, credence 0.97. Did not re-derive the explicit isomorphism symbolically (available transcription looked corrupted; disproportionate for importance 0.3), reflected in marginal_yield 0.15. (6) No dependents exist, so no steward notification; instead escalated to the Curator suggesting supports edges from the counterexample claims (085e13a9, a87e655a, 5b6a73ad) and a lateral link to the d=3-only claim (a63df1d9).
Assessed Verified
verdict confidence 0.90 · credence 0.97
The claim is the one non-obvious step in the geometric reading of the July 2026 counterexample to the Jacobian conjecture. That construction takes the covering map from P¹×Sym²(P¹), which is isomorphic to P¹×P², onto Sym³(P¹), identified with P³, sending a point and an unordered pair to the unordered triple; one removes the covering's ramification divisor together with the preimage of a hyperplane tangent, but not osculating, to the twisted cubic. The target complement is plainly affine three-space; the claim is that the source complement is as well, which is exactly what turns the restricted covering into a degree-three étale polynomial map from affine three-space to itself and hence into the counterexample. The isomorphism was described in the initial discussion as the main point in doubt, and it was then established several times over by independent methods: an explicit polynomial change of coordinates with a computed inverse, posted by Will Sawin from a ChatGPT-assisted calculation, and conceptual proofs by David Speyer, Jake Levinson, and Remy van Dobben de Bruyn, the last exhibiting the complement as a tower of Zariski A¹-bundles. Terence Tao's subsequent digestion of the counterexample reproves the trivialisation as part of a general bundle analysis. No source disputes the result. Its scope is narrow: the analogous construction yields an affine-space source only when the degree is three, which is why the counterexample lives specifically in three variables.
Claim entered the graph